Proving Lines are Perpendicular

Properties of Perpendicular LinesPerpendicular Lines Postulate:

• l1⊥l2 if and only if m1∙m2 = -1• That is, m2 = -1/m1,

The slopes are negative reciprocals of each other.

• Two non-vertical lines are perpendicular if and only if the product of their slopes is -1.Vertical and horizontal lines are perpendicular.

• In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other.

Theorem: Perpendicular to Parallel Lines:

and Then

• If two coplanar lines are each perpendicular to the same line, then they are parallel to each other.

Theorem: Two Perpendiculars:

Proof of Perpendicular to Parallel Lines Theorem

Statement Reason

1 l ll m, l ⊥ n Given2 ∠1 is a right angle Definition of lines⊥

3 m∠1 = 90o Definition of a right angle

4 m 2 ∠ = m∠1 Corresponding angles postulate

5 m∠2 = 90o Substitution property of equality

6 ∠2 is a right angle Definition of a right angle

7 m ⊥ n Definition of lines⊥

Given: l ll m and l ⊥ n Prove: m ⊥ n

Examples

1. Line r contains the points (-2,2) and (5,8). Line s contains the points (-8,7) and (-2,0). Is r ⊥ s?

2. Given the equation of line v isand line w is Is v ⊥ w?

Given the line

3.Find the equation of the line passing through ( 6,1) and perpendicular to the given line.

4. Find the equation of the line passing through ( 6,1) and parallel to the given line.

Homework

• Exercise 3.7 page 175: 1-35, odd.